📚 Year 11 OCR Statistics Formula & Theorem Quick Reference Handbook | Year 11 OCR 统计：公式定理速查手册
This quick reference handbook brings together all the essential formulae and theorems required for the OCR Year 11 Statistics syllabus. Each section presents key definitions and relationships with English and Chinese explanations side by side, helping you revise effectively and apply concepts confidently in exams.
本速查手册汇集了 OCR Year 11 统计课程要求的所有核心公式与定理。每个部分都以中英文对照的方式呈现关键定义和数量关系，帮助你高效复习，在考试中自信地运用各种概念。
1. Measures of Central Tendency | 集中趋势度量
The arithmetic mean for a dataset x₁, x₂, …, xₙ is calculated as x̄ = Σx / n. It is the most commonly used measure of central location but is sensitive to extreme values.
对于数据集 x₁, x₂, …, xₙ，算术平均值计算为 x̄ = Σx / n。这是最常用的集中位置度量，但对极端值敏感。
The median is the middle value when data are ordered. If n is even, the median is the average of the two central values. The median is resistant to outliers.
中位数是数据排序后处于中间位置的数值。当 n 为偶数时，中位数为居中的两个数值的平均。中位数抵抗异常值的影响。
The mode is the value with the highest frequency. A dataset can have no mode, one mode (unimodal), or more than one mode (multimodal).
众数是出现频率最高的值。一组数据可能没有众数、有一个众数（单峰）或多个众数（多峰）。
2. Measures of Spread | 离散程度度量
The range is the difference between the maximum and minimum values: Range = xmax − xmin. It is quick to compute but ignores the distribution of intermediate data.
极差是最大值与最小值之差：极差 = xmax − xmin。计算简便，但忽略了中间数据的分布情况。
Interquartile range (IQR) is Q₃ − Q₁, where Q₁ and Q₃ are the lower and upper quartiles. IQR captures the spread of the middle 50% of observations.
四分位距 (IQR) 为 Q₃ − Q₁，其中 Q₁ 和 Q₃ 分别是下四分位数和上四分位数。IQR 反映中间 50% 数据的离散程度。
Sample variance and standard deviation are given by:
s² = Σ (x − x̄)² / (n − 1)
and the standard deviation is s = √[ Σ(x − x̄)² / (n − 1) ]. For population data, divide by n instead of n − 1.
样本方差和标准差公式为：
s² = Σ (x − x̄)² / (n − 1)
标准差则为 s = √[ Σ(x − x̄)² / (n − 1) ]。对于总体数据，分母用 n 而非 n−1。
3. Probability Basics | 概率基础
For any event A, 0 ≤ P(A) ≤ 1. The complement rule states P(not A) = 1 − P(A). For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
对任何事件 A，有 0 ≤ P(A) ≤ 1。互补规则指出 P(非 A) = 1 − P(A)。对于互斥事件 A 和 B，P(A 或 B) = P(A) + P(B)。
For any two events A and B, the general addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Events A and B are independent if P(A ∩ B) = P(A) × P(B).
对于任意两个事件 A 和 B，一般加法规则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若 P(A ∩ B) = P(A) × P(B)，则事件 A 与 B 相互独立。
4. Tree Diagrams & Conditional Probability | 树状图与条件概率
Conditional probability is defined as P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0. Tree diagrams multiply probabilities along branches to find joint probabilities and sum across appropriate branches for ‘or’ probabilities.
条件概率定义为 P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。树状图通过沿分支相乘求联合概率，并在适当分支上求和计算“或”概率。
When using tree diagrams for successive events, remember that the sum of probabilities on branches from the same node equals 1. For conditional probabilities, the second set of branches changes according to the first outcome.
在使用树状图处理连续事件时，注意从同一节点出发的分支概率之和等于 1。对于条件概率，第二层分支会根据第一步结果而变化。
5. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable set of values x with probabilities P(X = x). The sum of all probabilities must be 1. The probability distribution can be displayed in a table.
离散随机变量 X 以概率 P(X = x) 取可数个值。所有概率之和必须为 1。该概率分布可用表格表示。
Expected value (mean) of X is E(X) = μ = Σ x · P(X = x). Variance is Var(X) = E[(X − μ)²] = Σ (x − μ)² P(X = x) = E(X²) − [E(X)]².
X 的期望值（均值）为 E(X) = μ = Σ x · P(X = x)。方差为 Var(X) = E[(X − μ)²] = Σ (x − μ)² P(X = x) = E(X²) − [E(X)]²。
6. Binomial Distribution | 二项分布
If X ~ B(n, p), then the probability of exactly k successes in n independent trials is:
P(X = k) = C(n, k) pk (1 − p)n−k
where C(n, k) = n! / [k! (n − k)!]. Each trial has only two outcomes (success/failure) and constant probability p.
若 X ~ B(n, p)，则在 n 次独立试验中恰好获得 k 次成功的概率为：
P(X = k) = C(n, k) pk (1 − p)n−k
其中 C(n, k) = n! / [k! (n − k)!]。每次试验只有两种结果（成功/失败）且概率 p 恒定。
The expected value and variance for a binomial distribution are E(X) = np and Var(X) = np(1 − p).
二项分布的期望值和方差为 E(X) = np，Var(X) = np(1 − p)。
7. Normal Distribution | 正态分布
A continuous random variable X that follows a normal distribution with mean μ and variance σ² is written as X ~ N(μ, σ²). The total area under the probability density curve equals 1.
当连续随机变量 X 服从均值为 μ、方差为 σ² 的正态分布时，记为 X ~ N(μ, σ²)。概率密度曲线下的总面积等于 1。
To find probabilities, standardise using:
Z = (X − μ) / σ
where Z ~ N(0, 1). Standard normal tables then give P(Z < z). For a range, compute P(a < X < b) = P( (a−μ)/σ < Z < (b−μ)/σ ).
求概率时，先标准化：
Z = (X − μ) / σ
其中 Z ~ N(0, 1)。然后查标准正态分布表得到 P(Z < z)。对于区间概率，计算 P(a < X < b) = P( (a−μ)/σ < Z < (b−μ)/σ )。
8. Scatter Diagrams & Correlation | 散点图与相关性
A scatter diagram plots bivariate data (x, y) to visually suggest whether there is a linear relationship. The direction can be positive (as x increases, y tends to increase), negative, or none.
散点图将双变量数据 (x, y) 绘制成图，直观显示是否存在线性关系。方向可以是正相关（x 增加时 y 趋于增加）、负相关或无相关。
Correlation measures the strength of a linear relationship but does not imply causation. Outliers can heavily influence the appearance of correlation.
相关性衡量线性关系的强度，但不意味着因果关系。异常值可能会极大地影响相关性的表现。
The product moment correlation coefficient (PMCC) is denoted by r, but for OCR Year 11 most emphasis is on interpreting given r-values or using Spearman’s rank.
积矩相关系数 (PMCC) 用 r 表示，但在 OCR Year 11 阶段更侧重解释给定的 r 值或使用斯皮尔曼等级相关系数。
9. Regression Line (Least Squares) | 回归直线（最小二乘法）
The equation of the regression line of y on x is y = a + bx. The slope b and intercept a are computed by:
b = Sxy / Sxx
a = ȳ − b x̄
where Sxy = Σxy − (Σx)(Σy)/n and Sxx = Σx² − (Σx)²/n.
y 对 x 的回归直线方程为 y = a + bx。斜率 b 和截距 a 由下式求得：
b = Sxy / Sxx
a = ȳ − b x̄
其中 Sxy = Σxy − (Σx)(Σy)/n，Sxx = Σx² − (Σx)²/n。
This line passes through the mean point (x̄, ȳ) and minimises the sum of squared vertical distances from the data points to the line.
该直线通过均值点 (x̄, ȳ)，并使数据点到直线的垂直距离平方和最小。
10. Spearman’s Rank Correlation | 斯皮尔曼等级相关系数
Spearman’s rank correlation coefficient, rs, measures the strength of monotonic association between two variables using their ranks. It is given by:
rs = 1 − (6 Σ d²) / [n(n² − 1)]
where d is the difference between the ranks of each pair, and n is the number of pairs.
斯皮尔曼等级相关系数 rs 利用等级数据来衡量两个变量之间单调关系的强度。公式为：
rs = 1 − (6 Σ d²) / [n(n² − 1)]
其中 d 是每对数据等级之差，n 为数据对数。
Values of rs range from −1 (perfect negative monotonic) to +1 (perfect positive monotonic). A value near 0 suggests no monotonic relationship.
rs 的取值范围从 −1（完全负单调关系）到 +1（完全正单调关系）。接近 0 的值表明没有单调关系。
11. Sampling & Bias | 抽样与偏差
A simple random sample gives every member of the population an equal chance of being selected. Systematic sampling selects every k-th element after a random start.
简单随机样本使总体中每个成员都有相等的被选中的机会。系统抽样则从一个随机起点开始，每隔 k 个元素抽取一个。
Stratified sampling divides the population into distinct groups (strata) and selects a random sample from each in proportion to its size, ensuring representation.
分层抽样将总体分成不同的组（层），并按各层大小比例从每层中随机抽样，以保证代表性。
Bias arises when a sample is not representative of the population. Common sources include voluntary response samples, convenience sampling, and poorly worded survey questions.
当样本不能代表总体时就会产生偏差。常见来源包括自愿应答样本、便利抽样以及问卷措辞不当。
12. Data Representation & Interpretation | 数据表示与解读
For grouped continuous data, frequency density is used in histograms: Frequency density = Frequency / Class width. The area of each bar is proportional to the frequency.
对于分组连续数据，直方图中使用频数密度：频数密度 = 频数 / 组距。每个直条的面积与频数成正比。
Cumulative frequency curves provide estimates of medians, quartiles, and percentiles. The median corresponds to a cumulative frequency of n/2, Q₁ to n/4, and Q₃ to 3n/4.
累积频数曲线可用于估计中位数、四分位数和百分位数。中位数对应的累积频数为 n/2，Q₁ 对应 n/4，Q₃ 对应 3n/4。
Box plots (box-and-whisker diagrams) display the minimum, Q₁, median, Q₃, and maximum. They effectively highlight skewness and potential outliers.
箱形图（箱须图）显示了最小值、Q₁、中位数、Q₃ 和最大值，能够有效突显分布偏态和潜在的异常值。
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